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II The Decision Proglem
I1 THE DECISION PROBLEM 1. Valuation in arbitrary domains. Universal validity In 5 2 of the preceding chapter we gave a definition of validity and satisfiability of wffs in finite domains. We now proceed to define the valuation of a w$ for a n arbitrary non-empty domain D , the elements of which may be of infinite number. This does not require any change in the relative definitions. The notions of propositional function and of an assignment for the free variables of a wff remain the same as before. The only difference is that the number of the n-adic propositional functions over D is infinite, if D has an infinite number of elements. Likewise, the valuation rules V l - V 6 remain unchanged as well as the notions of validity and satisfiability of a wff in a domain D. But now there may be an infinite number of assignments for individual variables and for predicate variables. Though the valuation of a wff is defined by the six rules ‘Vl-VVS, these rules in general do not afford a procedure by which the valuation may be effectively carried out. For by V6, the valuation of a wff for a given assignment is in general reduced to the valuation of another wff for an infinite number of assignments. A w$ i s called universally valid, if it i s valid in every non-empty domain. A w 8 is called satisfiable simply if there is a non-empty domain in which it i s satisfiable. The definitions of validity and satisfiability, together with the valuation rules V 4 and V6,immediately yield the following relations between these two concepts. If a wff % containing free variables is not valid in a domain D,3 is satisfiable in that domain, and with free variables is not universally valid, conversely. If a wff 3 is satisfiable, and vice versa. Since for wffs with free variables each of the two concepts of validity and satisfiability, as well as each of the concepts of
10
THE DECISION PROBLEM
universal validity and simple satisfiability, may be reduced to the other, it is sufficient to treat one of them. In the subsequent chapters we shall occupy ourselves only with the validity of wffs, though in some respects the concept of satisfiability is more easily handled when interpreting the wffs. The reason is that the semantic truth-definition for wffs we have given here may then be more conveniently brought into connection with a syntactical truthdefinition (cf. the preface). A syntactical truth-definition, which is not given in this book, is relative to a certain system of axioms. Such a system contains stipulations declaring a special class of wffs (the axioms) t o be universally valid. These stipulations further give prescriptions (deduction rules) for generating a new universally valid wff from a finite number of such wffs under certain conditions. A wff is provable in the system, if it is an axiom or if it may be generated from bhe set of axioms by a finite number of applications of the deduction rules. Syntactically true means provable. As the provable wffs are universally valid the notion of validity, but not that of satisfiability, has a direct connection with the syntactical truthdefinition. Even when using such a syntactical truth-definition a semantical truth-definition is not superfluous. For we must convince ourselves that the wffs called axioms are universally valid and that the deduction rules proceed from universally valid wffs to universally valid wffs, and that the wffs which are syntactically true really comprise all universally valid wffs. This can only be done by semantic methods. For the total number of wffs regarded here, there is no system of axioms delivering all universally valid wffs. Moreover, we are not only interested in the universal validity of wffs. Systems of axioms yielding all universally valid wffs exist for special classes of wffs. Wffs forming such a class are 1)the wffs of the propositional calculus, i.e. the wffs containing only propositional variables as prime formulas, 2) the wffs containing no predicate variables (the wffs of the pure calculus of equality), 3) the wffs containing no bound predicate variables (the wffs of the predicate calculus of
11
VALUATION IN ARBITRARY DOMAINS
first order) wherein the wffs containing the sign of equality may be included or not, and wffs of other special classes. We illustrate the notions of validity, satisfiability and universal validity by giving some examples.
ay)
(Ey)(Gx v This wff is universally valid; for if the element a of an arbitrary domain D is assigned to x and any monadic propositional function over D to G, the assignment a for y gives Gx v Gy the value T, since Gx and Gy get the same value. (1)
(2)
(EG)(x)(Ey)(Gxx& z y ) .
This formula is valid in a domain D which contains at least two elements; it is not valid in domains with only one element. If there is only one element in D ,Gxx & gets the value F by any assignment, since the elements assigned to x and y are the same. Consequently, the wff itself gets the value F. If there are at least two elements in D,we assign to G the dyadic propositional function over D the value of which is T only for an ordered pair of elements of D,the first and second member of which are the same. Then, given any assignment for x, an assignment for y can be found, namely an element of D different from the element assigned to x, so that Gxx&Ky gets the value T,which means that (EG)(x)(Ey)(Gxx& gets the value T.
6
G)
GX & Cy & X = Y . (3) This wff is valid in no domain at all; it is not even satisfiable. For if we assign to G an arbitrary monadic propositional function over a domain D and the elements a and b of D to x and y, either a and b are not the same elements, in which case x = y gets the value F ,or a and b are the same elements; in this case, Gx and Gy get the same value, so that either Gx or z y gets the value F. In & x = y gets the value F. any case Gx & (4)
(EG)((x)(Ey)Gxy& ( x ) G & ( x ) ( y ) ( z ) ( Gv
v Gxz))
This wff is valid in any domain having an infinite number of elements, and only in such a domain.
12
THE DECISION PROBLEM
If D has an infinite number of elements, there is a denumerably infinite subdomain of D called D, with the elements a,, a,, a,, .... We assign to G the dyadic propositional function over D the value of which is T only for the ordered pairs (un,a,,) (k> O ) and (6, uk)(k 3 l ) , b not being an element of D,.It is easy see that -toby this assignment (x)(Ey)Gxy & ( x ) G G & (z)(y)(z)(GxyvGyz vGxx) gets the value T, since each conjunction term gets the value T. On the other hand, if D has only a finite number of elements, let @ be an arbitrary dyadic propositional function over D . There is a subdomain of D having the following property $: If a is an arbitrary element of the subdomain and the ordered pair ( a , b ) gets the value T by @, then b is an element of the subdomain also. For instance, D itself is such a subdomain. Let Dlbe one of the subdomains with the property $ having the smallest - number of elements. If (x)(Ey)Gxy & (x)= & (x)(y)(x)(Gxyv Gyx v Gxx) is to get the value T by assigning @ to G and if c is some element of D,, there is another element el of D, not identical with e such that (c, cl) gets the value T by @, since (x)(Ey)Gxy & (x)= is to have the value T. Let D, be the set of all these elements cl. D, has a smaller number of elements than D,. If c2 is an arbitrary element of D, and if b is an element of D such that (c,, b ) gets the value T by @, the value -of (c, 6 ) for @ is T, since we are supposing that (~)(y)(z)(Gzy v Gyz v Gxz) gets the value T by assigning @ to G. Therefore, 6 is an element of D,. This means that D, also has the property '$. Since D, was supposed to be one of the smallest subdomains of D with the property '$, this is a contradiction.Therefore our supposition that the wff gets the value T by assigning @ to G, cannot be true. Since @ was arbitrary, our wff (4) gets the value F in any finite domain.
2. Equivalences If a wff $ ++ ? 23, I i.e. v 23 & 8 v % is valid in a domain D, the valuations of 8 and of 23 are the same for two relative assignments which coincide with respect t o the variables common to % and B. For it is an immediate consequence of the valuation rules V4 and V 5 that by any assignment rzI tf 23 gets the value T, if and only
a
13
EQUIVALENCES
if both '21 and % get the value T by this assignment, or if both wffs get the value F. If '21 ++ '23 is valid in D,if (21 is part of a wff C5 and if the wff & is obtained from 6 by replacing the part '21 of 0: by $'3 and by dropping, if necessary, vacuous quantifiers (those quantifiers the scopes of which do not contain the variable of the quantifier), C5 ++Elis valid in D, provided that contains no free variables not occurring in '21. For it is clear that any assignment gives the same value to 6 and El. '21 and % are called equivalent wfjs, if '21 ++ 93 is universally valid. If 8 and 2 '3 are equivalent wffs and if 6 and have the same meaning as before, 6 and El are equivalent wffs. Equivalent wffs are valid or not valid in exactly the same domains. In the following we state several equivalences which are used later on. E l . From a wff % we get an equivalent wff $' 3 by the following transformation. The variable of a quantifier of 8 and the same variable in all places of the scope of the quantifier is replaced by another variable of the same character. This transformation is restricted to the case in which !8 is a wff and the newly introduced variable is not identical with one of the variables occurring in the scope of the quantifier. Example: (Ex)Gx & (Ex)Gx may be transformed into (Ez)Gx& & (Ey)Gy, and conversely. Indeed, it is seen by V 6 that by any assignment for the free variables of '21 in any domain the part of '21 consisting of the quantifier and its scope gets the same value T or F before and after the transformation, as does %. If a wff (21 is transformed in the described way one or more times we shall say that we have rewritten the bound variables of 8. Subsequently we shall not make much use of the syntactical variables a, b, C, ..., U, %, 2 ' 33, ..., and U", %", iBn,..., which stand respectively for any individual variable, any propositional variable and for any n-adic predicate variable. A statement, for instance, concerning the validity of some wff (z)(Ey)(z)%(x,y, x ) equally m *-
14
THE DECISION PROBLEM
holds for any wff (a)(Eb)(c)%(a, b, c) resulting from (x)(Ey)(z)%(x,y,z) by rewriting the bound variables. Therefore it is sufficient to state such statements for specially chosen bound variables. In stating the equivalences E 5 - E l 6 a may designate individual variables as well as propositional variables and predicate variables, since these equivalences hold for all kinds of variables. Pairs of equivalent wffs are the following ones: E2.
E3.
%%%
E4.
(m)
E5. E6.
E7. E8E9. E10. Ell. E12. E13. E14. El5. E16.
(m) (Ea)(%(a)v %(a))
%(a)) (Ea)%(a)& % % & (Ea)%(a) (Ea)%(a)v % B v (Ea)%(a) (a)(%(a)
(a)%(a) & B B & (a)%(a)
(a)%(a) v % % v(a)%(a) E17. ( x ) ( x =v~%(x, y)) E18. (Ex)(x=y & %(x, y))
and and and and and and and and and and and and and and and and and
The equivalences E2 - E l 6 are an immediate consequence of the valuation rules V l - - 6 . I n E 9 - E l 6 , the wff B of course cannot contain the variable a, because we have assumed that we have a pair of wffs. I n El7 and E18, %(y, y) is the wff resulting from %(x, y) by substituting y for x in all places where x occurs. The pairs of wffs mentioned in El7 and E l 8 are recognized as pairs of equivalent wffs as follows. If we have an assignment for the free variables of %(y,y) in a domain D and an assignment for %(x, y) which is the same as that for %(y, y) for the variables different from x, and by which the same element of D is assigned to x and to y, then these assignments give the same value to %(y, y) and to
#-FORMULAS
15
X(x, y). This is clear for prime formulas and may be proven for other wffs by induction, according to the construction of wffs by W1-W6 of Chapter I, 9 1. If an assignment €or the free variables of %(y, y) is given, this assignment gives the same value t o (Ex)(x=y & %(x, y)). For if X(y, y) gets the value T by this assignment, so does x = y & %(x, y), provided t’he assignment is completed by assigning to 2 the same element as to y, If X(y, y) gets the value F by this assignment, x = y & %(x, y) also receives the value P by any assignment coinciding with the assignment for X(y, y) with respect to the variables different from x. This results from what we have said above if the same element is assigned to x and y. If different elements are assigned to x and to y, x= y and therefore x = y & 8 ( x ,y) get thevalue F . I n a similar manner it is to be seen that %(y, y) and (z)(x=y v %(x, y)) are equivalent wffs. v ... v 8, is transformed into an equiE19. A disjunction valent wff if the succession of the disjunction terms is somehow altered. The analogous statement is true for a conjunction. E20. The wffs X and X & ‘23 are equivalent if ‘23 is universally valid. The wffs X and % v 23 are equivalent, if 23 is not satisfiable is universally valid. in any domain, i.e. if E21. The wffs X v (23 & C?) and % v ‘23 & % v 6 as well as the wffs 8 & ‘23 v 6 and (8 & ‘23) v (X & 6)are equivalent wffs. E19--E21 are easily recognized to be true by the valuation rules.
3. #-formulas. Elimination of the propositional variables We often have to form sentences like this: “A wff gets the value T or F if the element a is assigned to x, the element b to y, the monadic propositional function Y to GI, etc.” It is desirable to have a shorter form for such statements; this will be obtained as follows. Instead of saying “ x = y & (z)Gzz gets the value T ( F ) , if we assign a to x, b to y and 0 to G”, we use in a synonymous manner the sentence”, a=b & (z)@az is true (false)”. Likewise, “X v Y & (G)(Gx v Gy) gets the value F , if we assign T to X , F to Y , a to x, and a t o y” and “T v F & (G)(GavGa) is false” shall have the same sense. By this we do not intend to extend the notion of wff. Formal expressions like T v F & (G)(Ga v Ga) only
16
THE DECISION PROBLEM
occur in statements about wffs; they do not belong to our formal language. We shall call such formal expressions S-formulas. S-formulas are those, and only those, formal expressions which result from a wff with free variables by replacing all or some of these variables by the names of things assigned to them in a domain. The replacement of a free variable must be done in all places where it occurs. Wffs without free variables may also occur among the S-formulas. If an S-formula results from a wff in this way, we say that the 8-formula belongs to the wff and the assignment for the free variables of the wff. The wff and the assignment to which an S-formula belongs, is not uniquely determined by the S-formula, even if we disregard the fact that different letters may be used for the free variables of the wff. E.g. a=a belongs to x = x and the assignment a for x, but also to x= y and the assignment a for x and a for y. T v & Z belongs to X v & Z and the assignment T for X , but also to Y vx $ 2 and the assignment T for X and T for Y (but not to 2 v z & 2). The same S-formula may belong to a wff % and an assignment Ql and to a wff 23 and the assignment Q, which is relative to the same domain as Ql. % and 93 can only differ in such a way that the place for a variable in B is occupied by another variable of the same character in 23. Variables in corresponding places of B and ‘B get, if any, the same assignments by Q, and Q2. If two places are occupied by the same variable in one wff and the corresponding places in the other wff by different variables, then the assignment for the two variables of the second wff is the same. Q, and Q2 give the same value to % and to 23 if they are complete assignments and otherwise, if their completion is achieved by adding the same assignments for the other free variables. This is seen at once for prime formulas by inspecting the valuation rules 8 1 - V 3 and is proved for other wffs by induction according to 84- 8 6 . Thus the statement “An S-formula is true in a domain D” has sense without naming a special wff to which the S-formula belongs. “An S-formula without free variables is true (false) in a domain D” is only another expression for “The S-formula belongs to a wff and an assignment for its free variables over D which gives the wff the value T(F)”, or
x
S -FORMULAS
17
“Any wff and pertaining assignment gives the wff the value T ( F )in D if the X-formula belongs to them”. “An 8-formula with free variables is true (false) in D for an assignment for the free vari-
ables” means “The 8-formula belongs to a wff and an assignment for a part of the free variables of the wff over D which, completed by assigning to the other free variables the same things as to the free variables of the X-formula identical with them, gives the wff the value T ( F ) in D”.Note that not for every assignment for the free variables of an X-formula in an arbitrary domain the formula, is true or false, since this assignment must be made up to form a complete assignment with the assignment to which the X-formula belongs. The X-formula Yxx,for instance, can be true or false only if an element of the domain D over which Y i s defined, is assigned to x. a=x & Gax is true or false only for assignments for x and G in domains of which a is an element. Two 8-formulas Gland Gz are called equivalent if Gltf G2is true in every domain for all assignments for the free variables of G1 tf Gz. For wffs this notion of equivalence is the same as that given in $ 2 of this chapter. G3and G4 are equivalent 8-formulas if the following conditions are fulfilled: GI and G2are equivalent 8-formulas; Gz does not contain any free variables not occurring the X-formula G4results from in Gl;G1is part of an S-formula G3; G3 by replacing the part Gl of G3 by Gz and by dropping, if necessary, quantifiers which have become superfluous. If a wff is equivalent to the X-formula T ( F ) , then the wff is universally valid (valid in no domain at all). As i s seen by the definition of equivalence, this notion i s restricted to X-formulas containing only T and F besides the primitive symbols of wgs. Such 8-formulas can be transformed into equivalent wffs or into one of the two 8-formulas, T or F . This is done by replacing in the S-formula, beginning anywhere, T by F , by T , T v and a v T by T , F v 8 and ’% v P by a, T & rll and ‘% & T by a, F & % and & F by F and by simultaneously dropping vacuous quantifiers, if necessary. Since all the mentioned pairs of S-formulas are pairs of equivalent 8-formulas, the 8-formula is transformed into an equivalent one. Since each reduction step diminishes the number
18
THE DECISION PROBLEM
of logical constants, this reduction can be carried on until T and F have disappeared from the S-formula, or until the S-formula is reduced to T or to F . This may be used to show that we need not investigate the validity or satisfiability of wffs containing propositional variables, because the propositional variables can be eliminated from a wff. Validity or satisfiability of a wff always means the validity of a certain wff containing no free variables. A WTJ with bound propositional variables may always be repluced by a n equivalent w g containing no propositional variables, or it m y be recognized at once as universally valid or valid in no domain at all. Each such wff can at first be transformed into an equivalent S-formula without bound propositional variables by replacing in the wff, beginning at an arbitrary point, every part ( X ) % ( X )by %(T)& %(F) and every part ( E X ) % ( X ) by %(T)v %(F). This S-formula is then reduced in the way described above. If the formula resulting in the end is T or F , the wff is universally valid or valid in no domain at all; otherwise, we get an equivalent wff without propositional variables. The well-known truth-table decision procedure for wffs of the propositional calculus, i.e. for wffs containing no individual and no predicate variables, is included herein. As an example we consider the wff:
( X ) ( Xv ( y ) ( F yv 15)) & ( E Y ) ( Y& (EG)(Ex)(Gz& Y ) ) . We first replace
( X ) ( X v (Y)(y=Y
" m by T v ( Y ) ( y = y v m
-
F v (Y)(Y=Y+
F)
which reduces to T,so that the wff itself is reduced to
(EY)(Y & (EG)(Ez)(Gz& Y ) ) . We replace this wff by
(T & (EG)(Ez)(Gx& T))v ( F & (EG)(Ez)(Gz& P ) ) which reduces to (EG)(Ex)Gx.(EG)(Ex)Gxis equivalent to the wff from which we were starting.
19
NORMAL FORMS
4.
Normal forms For this section we need the concept of P-constituent of a wff defined in 9 1 of Chapter I. A wff is said to express a tautology, or to be tautologous, if every S-formula reduces to T which results from the wff by replacing the P-constituents in an arbitrary way by T or F , but so that equally shaped P-constituents are replaced by the same letter. A tautologous wff is of course universally valid, but not every universally valid wff is tautologous. E.g. (z)(z= x) and (z)(Gzv&) are universally valid, but not tautologous. Two tautologous wffs are equivalent. Every wff can be brought into certain normal forms by means of equivalence transformations. There are several normal forms. A wff is said to be in conjunctive normal form if it is a disjunction the terms of which are P-constituents or the negations of P-constituents of the wff or if it is a conjunction of such disjunctions. Wffs which consist only of one P-constituent or of the negation of a P-constituent shall be included among the above disjunctions. We first show that for every non-tautologous wff an equivalent wff in conjunctive normal form can be constructed. a1,212, ..., am may be the P-constituents of the wff, P-constituents equally shaped not being counted twice. We will designate the wff by B(a1,..., a,). B(T, a2,..., a,) and B(F,212, ..., a,) are the #-formulas resulting from B(2Xl, ..., am)by replacing 2I1 everywhere by T ( F ) . B('2I1,..., a,) is equivalent to the S-formula
B(T,a2,..., %,). gets the value F , then the S-formula
a1v B(F, 212, ..., a,)
&,
v
For if by any assignment gets the same value by this assignment as
B v B ( F , a2,..., %), tk v B(T, a2,..., %,). This formula reduces to B(F, 212, ..., a,) which gets the same value
%(al,
as ..., a,). If by any assignment S-formula gets the same value as
a1 gets
the value T,the
T v B ( F , g2,..., '3%) & ?I v B(T,'?I2, ..., a,), which reduces to %(T,S2,..., a,).
20
THE DECISION PROBLEM
We shall prove our theorem by induction on the number n of P-constituents. If this number is one, %(9ll) is equivalent to v % ( F )& v %(T).We eliminate T and F in this 8-formula or 211 & % as conby the methods of 5 3 and get either 2117 or junctive normal form, since B(F)and %(T) cannot both reduce to T ;otherwise %(211) would be tautologous. Let us assume that we have shown our theorem to be true for a wff with less than n P-constituents. ..., a,) is equivalent to
a1
%(al,
211 v % ( F ,a2,..., a,) &
v
%(T,a2,...)am)
I n this S-formula F and T are again eliminated by the methods .of 0 3. The result is a wff of one of the following forms: 2l1v6&l,v%, 8 1 83%
v 9,
211v&&q, v 9,
a l V &
a1 &Z1, %I, &.
The wff cannot reduce to F , as is seen by its structure; it cannot reduce t o T,for in that case both %(F,a2,..., am) and %(T,g2,...,3%) would reduce to T,which is impossible since a2,..., is not tautologous. The last three of the above eight wffs are in conjunctive normal form. I n the other five wffs, 6 and 3 are wffs with only the P-constituents az7 ..., which according to our assumption can be brought into conjunctive normal forms 6’ and 9’. Because ‘ill1v G’ and v 9’can be transformed into a conjunctive normal form by means of the first equivalence E 2 1 of 5 2 , this holds too for the five wffs. This completes our induction. A tautologous wff with the P-constituents a17 ...)‘i&can also be given a conjunctive normal form; e.g. 3, v is such a form. A wff is in disjunctive normal form if it is a conjunction of P-constituents and the negations of P-constituents or if it is a disjunction of such conjunctions. Wffs consisting only of one P-constituent or of the negation of a P-constituent shall be included among the above conjunctions. I n order to show that a wff a can always be brought into a n equivalent disjunctive normal form, we fist construct a conjunctive normal form B1& ... & Brn of 8. According to E 2 of 5 2, 2l is equivalent to and therefore to B1& ... & 58%. Using E3,
%(al,
al
an)
21
NORMAL FORMS
Em.
is v ... v By E4 every !?jl& ... & !Bmis transformed into transformed into a conjunction whose terms are negations of P-constituents or double negated P-constituents. By eliminating the double negations according to E2, we get a disjunctive normal form of '21. Every wfl can be given an equivalent normal form in which the negation-bar stands only over prime formulas. If a wff is given, by successive applications of E3, E 4 , E5 and E6 the negation signs can be brought farther and farther inside until finally they stand, one or more times, only over the prime formulas. By means of E2 the negation signs over the prime formulas are removed until there is at most one left. Example: The wff
(G)(Ex)(Ey)(Gxy &2
7 )
becomes (EG)(x)(y)(= v x= y )
by the transformation. We m y further transform any w 8 into an equivalent one in which the quantifiers occur only at the beginning with scopes extending to the end of the w4, no quantifier standing under a negation-bar. We first transform the given wff into one in which the negationbars stand a t most over prime formulas. Then we rewrite the bound variables (see E l of 3 2 ) in a way such that all quantifiers have different variables. Subsequently, we place all the quantifiers at the beginning of the wff with scopes extending to the end of the wff, by using several times the equivalences E9-El6 of 5 2. This generally can be done in more than one way. For instance, the quantifiers can be placed a t the beginning of the wff in the order in which they occur, but also any succession of the initially placed quantifiers can be obtained such that any quantifier which was in the scope of another quantifier remains in the scope of that quantifier. A wff in which all the quantifiers are placed at the beginning of the wff with scopes extending to the end of the wff is said t o be in a prenex normal form. The succession of quantifiers with which the wff begins is called the prefix of the wff. The part of the wff which remains if all the quantifiers are deleted is called the matrix of the wff.
22
TI333 DECISION PROBLEM
Example: A prenex normal form of (x)(Ey)Gxy& (Ez)Hz is ( E x ) ( y ) ( E z ) ( K& y H z ) with the prefix ( E x ) ( y ) ( E zand ) the matrix & Hz. Other prenex normal forms are ( E x ) ( E z ) ( y ) ( K & y Hz) and ( E z ) ( E x ) ( y ) ( G& Hz). 5.
Three forms of the decision problem If a wff is valid (satisfiable) in a domain D, it is also valid (satisfiable) in a domain D, which has the same cardinal number as D,i.e. the elements of which may be put into one-to-one correspondence with those of D. Indeed, through the assignment used for the valuation of a wff in D,there may be determined in an unambigous way assignments relative to D, which give the same valuation. If a is assigned to an individual variable relative to D , the corresponding element a, of D, is used for the assignment relative to D,. If the n-adic propositional function rP is assigned to U" in D,the assignment for Unrelative to D,is the propositional function rP1 over Dlwhich assigns the value T to those, and only those, n-tuples (b,, ..., b,) over D,for which rP gives the value T to the corresponding n-tuples (a,, ..., a,) over D , a, corresponding t o bi. Therefore, disregarding those wfla which are universally valid (satisfiable in every non-empty domain) and disregarding those which have this property in no domain, the assertion of the validity (or satisfiability) of a w@ in a domain D is equivalent to a statement about the cardinal number of D. The definition of validity does not supply us with a criterion for recognizing in the general case whether a given wff is valid (satisfiable) in a given domain, or whether it is universally valid (satisfiable) at all. The problem of finding an effective procedure for determining the validity (satisfiability) of a given wff is called the decision problem. More precisely, as the validity and satisfiability of a wf€ depend on the cardinal number of the relative domain, the decision problem may be stated in several distinct forms. We formulate all of them with respect t o the problem of validity, since this is sufficient (see 8 1 of this chapter). The most important form is this:
THREE FORMS OF THE DECISION PROBLEM
23
I. T o decide for a given wfl whether it i s universally valid or not. A more comprising form of the problem is this: 11. To decide for a given wfl whether it is universally valid. If it i s not universally valid, to decide whether it i s valid in no domain whatsoever or in some domain. I n the last case the cardinal numbers of the domains for which it i s valid are to be determined. Other forms of the decision problem may be stated as well, e.g. the following one: 111. T o decide for a given wfl whether it i s valid in all domains with a finite number of elements or not. As far as the literature goes nearly all investigations about soluble cases of the decision problem concern problems I and 11; I11 has been dealt with only in connection with 11,since a solution of I1 includes a solution of I11 (but see Q 2 of chapter VIII). Of course it would be most desirable to have a decision procedure applicable to all wffs. Investigations made at first by A. Church [l, 21, then by A. Turing [l] and others, by which the somewhat vague intuitive notion of a decision procedure was replaced by a precise definition, show that a general solution of the decision problem in the forms I and I1 cannot be found. B. A. Trachtenbrod [l] has shown also that a general solution of the form I11 of the decision problem is not possible. To enter into the particulars of these investigations is outside the limits of this book. It remains t o look for a solution of the problem for special cases, i.e. for certain special well-defined classes of wffs. I n the following we give a survey of the more important special cases for which a solution has hitherto been reached.
10
THE DECISION PROBLEM
universal validity and simple satisfiability, may be reduced to the other, it is sufficient to treat one of them. In the subsequent chapters we shall occupy ourselves only with the validity of wffs, though in some respects the concept of satisfiability is more easily handled when interpreting the wffs. The reason is that the semantic truth-definition for wffs we have given here may then be more conveniently brought into connection with a syntactical truthdefinition (cf. the preface). A syntactical truth-definition, which is not given in this book, is relative to a certain system of axioms. Such a system contains stipulations declaring a special class of wffs (the axioms) t o be universally valid. These stipulations further give prescriptions (deduction rules) for generating a new universally valid wff from a finite number of such wffs under certain conditions. A wff is provable in the system, if it is an axiom or if it may be generated from bhe set of axioms by a finite number of applications of the deduction rules. Syntactically true means provable. As the provable wffs are universally valid the notion of validity, but not that of satisfiability, has a direct connection with the syntactical truthdefinition. Even when using such a syntactical truth-definition a semantical truth-definition is not superfluous. For we must convince ourselves that the wffs called axioms are universally valid and that the deduction rules proceed from universally valid wffs to universally valid wffs, and that the wffs which are syntactically true really comprise all universally valid wffs. This can only be done by semantic methods. For the total number of wffs regarded here, there is no system of axioms delivering all universally valid wffs. Moreover, we are not only interested in the universal validity of wffs. Systems of axioms yielding all universally valid wffs exist for special classes of wffs. Wffs forming such a class are 1)the wffs of the propositional calculus, i.e. the wffs containing only propositional variables as prime formulas, 2) the wffs containing no predicate variables (the wffs of the pure calculus of equality), 3) the wffs containing no bound predicate variables (the wffs of the predicate calculus of
11
VALUATION IN ARBITRARY DOMAINS
first order) wherein the wffs containing the sign of equality may be included or not, and wffs of other special classes. We illustrate the notions of validity, satisfiability and universal validity by giving some examples.
ay)
(Ey)(Gx v This wff is universally valid; for if the element a of an arbitrary domain D is assigned to x and any monadic propositional function over D to G, the assignment a for y gives Gx v Gy the value T, since Gx and Gy get the same value. (1)
(2)
(EG)(x)(Ey)(Gxx& z y ) .
This formula is valid in a domain D which contains at least two elements; it is not valid in domains with only one element. If there is only one element in D ,Gxx & gets the value F by any assignment, since the elements assigned to x and y are the same. Consequently, the wff itself gets the value F. If there are at least two elements in D,we assign to G the dyadic propositional function over D the value of which is T only for an ordered pair of elements of D,the first and second member of which are the same. Then, given any assignment for x, an assignment for y can be found, namely an element of D different from the element assigned to x, so that Gxx&Ky gets the value T,which means that (EG)(x)(Ey)(Gxx& gets the value T.
6
G)
GX & Cy & X = Y . (3) This wff is valid in no domain at all; it is not even satisfiable. For if we assign to G an arbitrary monadic propositional function over a domain D and the elements a and b of D to x and y, either a and b are not the same elements, in which case x = y gets the value F ,or a and b are the same elements; in this case, Gx and Gy get the same value, so that either Gx or z y gets the value F. In & x = y gets the value F. any case Gx & (4)
(EG)((x)(Ey)Gxy& ( x ) G & ( x ) ( y ) ( z ) ( Gv
v Gxz))
This wff is valid in any domain having an infinite number of elements, and only in such a domain.
12
THE DECISION PROBLEM
If D has an infinite number of elements, there is a denumerably infinite subdomain of D called D, with the elements a,, a,, a,, .... We assign to G the dyadic propositional function over D the value of which is T only for the ordered pairs (un,a,,) (k> O ) and (6, uk)(k 3 l ) , b not being an element of D,.It is easy see that -toby this assignment (x)(Ey)Gxy & ( x ) G G & (z)(y)(z)(GxyvGyz vGxx) gets the value T, since each conjunction term gets the value T. On the other hand, if D has only a finite number of elements, let @ be an arbitrary dyadic propositional function over D . There is a subdomain of D having the following property $: If a is an arbitrary element of the subdomain and the ordered pair ( a , b ) gets the value T by @, then b is an element of the subdomain also. For instance, D itself is such a subdomain. Let Dlbe one of the subdomains with the property $ having the smallest - number of elements. If (x)(Ey)Gxy & (x)= & (x)(y)(x)(Gxyv Gyx v Gxx) is to get the value T by assigning @ to G and if c is some element of D,, there is another element el of D, not identical with e such that (c, cl) gets the value T by @, since (x)(Ey)Gxy & (x)= is to have the value T. Let D, be the set of all these elements cl. D, has a smaller number of elements than D,. If c2 is an arbitrary element of D, and if b is an element of D such that (c,, b ) gets the value T by @, the value -of (c, 6 ) for @ is T, since we are supposing that (~)(y)(z)(Gzy v Gyz v Gxz) gets the value T by assigning @ to G. Therefore, 6 is an element of D,. This means that D, also has the property '$. Since D, was supposed to be one of the smallest subdomains of D with the property '$, this is a contradiction.Therefore our supposition that the wff gets the value T by assigning @ to G, cannot be true. Since @ was arbitrary, our wff (4) gets the value F in any finite domain.
2. Equivalences If a wff $ ++ ? 23, I i.e. v 23 & 8 v % is valid in a domain D, the valuations of 8 and of 23 are the same for two relative assignments which coincide with respect t o the variables common to % and B. For it is an immediate consequence of the valuation rules V4 and V 5 that by any assignment rzI tf 23 gets the value T, if and only
a
13
EQUIVALENCES
if both '21 and % get the value T by this assignment, or if both wffs get the value F. If '21 ++ '23 is valid in D,if (21 is part of a wff C5 and if the wff & is obtained from 6 by replacing the part '21 of 0: by $'3 and by dropping, if necessary, vacuous quantifiers (those quantifiers the scopes of which do not contain the variable of the quantifier), C5 ++Elis valid in D, provided that contains no free variables not occurring in '21. For it is clear that any assignment gives the same value to 6 and El. '21 and % are called equivalent wfjs, if '21 ++ 93 is universally valid. If 8 and 2 '3 are equivalent wffs and if 6 and have the same meaning as before, 6 and El are equivalent wffs. Equivalent wffs are valid or not valid in exactly the same domains. In the following we state several equivalences which are used later on. E l . From a wff % we get an equivalent wff $' 3 by the following transformation. The variable of a quantifier of 8 and the same variable in all places of the scope of the quantifier is replaced by another variable of the same character. This transformation is restricted to the case in which !8 is a wff and the newly introduced variable is not identical with one of the variables occurring in the scope of the quantifier. Example: (Ex)Gx & (Ex)Gx may be transformed into (Ez)Gx& & (Ey)Gy, and conversely. Indeed, it is seen by V 6 that by any assignment for the free variables of '21 in any domain the part of '21 consisting of the quantifier and its scope gets the same value T or F before and after the transformation, as does %. If a wff (21 is transformed in the described way one or more times we shall say that we have rewritten the bound variables of 8. Subsequently we shall not make much use of the syntactical variables a, b, C, ..., U, %, 2 ' 33, ..., and U", %", iBn,..., which stand respectively for any individual variable, any propositional variable and for any n-adic predicate variable. A statement, for instance, concerning the validity of some wff (z)(Ey)(z)%(x,y, x ) equally m *-
14
THE DECISION PROBLEM
holds for any wff (a)(Eb)(c)%(a, b, c) resulting from (x)(Ey)(z)%(x,y,z) by rewriting the bound variables. Therefore it is sufficient to state such statements for specially chosen bound variables. In stating the equivalences E 5 - E l 6 a may designate individual variables as well as propositional variables and predicate variables, since these equivalences hold for all kinds of variables. Pairs of equivalent wffs are the following ones: E2.
E3.
%%%
E4.
(m)
E5. E6.
E7. E8E9. E10. Ell. E12. E13. E14. El5. E16.
(m) (Ea)(%(a)v %(a))
%(a)) (Ea)%(a)& % % & (Ea)%(a) (Ea)%(a)v % B v (Ea)%(a) (a)(%(a)
(a)%(a) & B B & (a)%(a)
(a)%(a) v % % v(a)%(a) E17. ( x ) ( x =v~%(x, y)) E18. (Ex)(x=y & %(x, y))
and and and and and and and and and and and and and and and and and
The equivalences E2 - E l 6 are an immediate consequence of the valuation rules V l - - 6 . I n E 9 - E l 6 , the wff B of course cannot contain the variable a, because we have assumed that we have a pair of wffs. I n El7 and E18, %(y, y) is the wff resulting from %(x, y) by substituting y for x in all places where x occurs. The pairs of wffs mentioned in El7 and E l 8 are recognized as pairs of equivalent wffs as follows. If we have an assignment for the free variables of %(y,y) in a domain D and an assignment for %(x, y) which is the same as that for %(y, y) for the variables different from x, and by which the same element of D is assigned to x and to y, then these assignments give the same value to %(y, y) and to
#-FORMULAS
15
X(x, y). This is clear for prime formulas and may be proven for other wffs by induction, according to the construction of wffs by W1-W6 of Chapter I, 9 1. If an assignment €or the free variables of %(y, y) is given, this assignment gives the same value t o (Ex)(x=y & %(x, y)). For if X(y, y) gets the value T by this assignment, so does x = y & %(x, y), provided t’he assignment is completed by assigning to 2 the same element as to y, If X(y, y) gets the value F by this assignment, x = y & %(x, y) also receives the value P by any assignment coinciding with the assignment for X(y, y) with respect to the variables different from x. This results from what we have said above if the same element is assigned to x and y. If different elements are assigned to x and to y, x= y and therefore x = y & 8 ( x ,y) get thevalue F . I n a similar manner it is to be seen that %(y, y) and (z)(x=y v %(x, y)) are equivalent wffs. v ... v 8, is transformed into an equiE19. A disjunction valent wff if the succession of the disjunction terms is somehow altered. The analogous statement is true for a conjunction. E20. The wffs X and X & ‘23 are equivalent if ‘23 is universally valid. The wffs X and % v 23 are equivalent, if 23 is not satisfiable is universally valid. in any domain, i.e. if E21. The wffs X v (23 & C?) and % v ‘23 & % v 6 as well as the wffs 8 & ‘23 v 6 and (8 & ‘23) v (X & 6)are equivalent wffs. E19--E21 are easily recognized to be true by the valuation rules.
3. #-formulas. Elimination of the propositional variables We often have to form sentences like this: “A wff gets the value T or F if the element a is assigned to x, the element b to y, the monadic propositional function Y to GI, etc.” It is desirable to have a shorter form for such statements; this will be obtained as follows. Instead of saying “ x = y & (z)Gzz gets the value T ( F ) , if we assign a to x, b to y and 0 to G”, we use in a synonymous manner the sentence”, a=b & (z)@az is true (false)”. Likewise, “X v Y & (G)(Gx v Gy) gets the value F , if we assign T to X , F to Y , a to x, and a t o y” and “T v F & (G)(GavGa) is false” shall have the same sense. By this we do not intend to extend the notion of wff. Formal expressions like T v F & (G)(Ga v Ga) only
16
THE DECISION PROBLEM
occur in statements about wffs; they do not belong to our formal language. We shall call such formal expressions S-formulas. S-formulas are those, and only those, formal expressions which result from a wff with free variables by replacing all or some of these variables by the names of things assigned to them in a domain. The replacement of a free variable must be done in all places where it occurs. Wffs without free variables may also occur among the S-formulas. If an S-formula results from a wff in this way, we say that the 8-formula belongs to the wff and the assignment for the free variables of the wff. The wff and the assignment to which an S-formula belongs, is not uniquely determined by the S-formula, even if we disregard the fact that different letters may be used for the free variables of the wff. E.g. a=a belongs to x = x and the assignment a for x, but also to x= y and the assignment a for x and a for y. T v & Z belongs to X v & Z and the assignment T for X , but also to Y vx $ 2 and the assignment T for X and T for Y (but not to 2 v z & 2). The same S-formula may belong to a wff % and an assignment Ql and to a wff 23 and the assignment Q, which is relative to the same domain as Ql. % and 93 can only differ in such a way that the place for a variable in B is occupied by another variable of the same character in 23. Variables in corresponding places of B and ‘B get, if any, the same assignments by Q, and Q2. If two places are occupied by the same variable in one wff and the corresponding places in the other wff by different variables, then the assignment for the two variables of the second wff is the same. Q, and Q2 give the same value to % and to 23 if they are complete assignments and otherwise, if their completion is achieved by adding the same assignments for the other free variables. This is seen at once for prime formulas by inspecting the valuation rules 8 1 - V 3 and is proved for other wffs by induction according to 84- 8 6 . Thus the statement “An S-formula is true in a domain D” has sense without naming a special wff to which the S-formula belongs. “An S-formula without free variables is true (false) in a domain D” is only another expression for “The S-formula belongs to a wff and an assignment for its free variables over D which gives the wff the value T(F)”, or
x
S -FORMULAS
17
“Any wff and pertaining assignment gives the wff the value T ( F )in D if the X-formula belongs to them”. “An 8-formula with free variables is true (false) in D for an assignment for the free vari-
ables” means “The 8-formula belongs to a wff and an assignment for a part of the free variables of the wff over D which, completed by assigning to the other free variables the same things as to the free variables of the X-formula identical with them, gives the wff the value T ( F ) in D”.Note that not for every assignment for the free variables of an X-formula in an arbitrary domain the formula, is true or false, since this assignment must be made up to form a complete assignment with the assignment to which the X-formula belongs. The X-formula Yxx,for instance, can be true or false only if an element of the domain D over which Y i s defined, is assigned to x. a=x & Gax is true or false only for assignments for x and G in domains of which a is an element. Two 8-formulas Gland Gz are called equivalent if Gltf G2is true in every domain for all assignments for the free variables of G1 tf Gz. For wffs this notion of equivalence is the same as that given in $ 2 of this chapter. G3and G4 are equivalent 8-formulas if the following conditions are fulfilled: GI and G2are equivalent 8-formulas; Gz does not contain any free variables not occurring the X-formula G4results from in Gl;G1is part of an S-formula G3; G3 by replacing the part Gl of G3 by Gz and by dropping, if necessary, quantifiers which have become superfluous. If a wff is equivalent to the X-formula T ( F ) , then the wff is universally valid (valid in no domain at all). As i s seen by the definition of equivalence, this notion i s restricted to X-formulas containing only T and F besides the primitive symbols of wgs. Such 8-formulas can be transformed into equivalent wffs or into one of the two 8-formulas, T or F . This is done by replacing in the S-formula, beginning anywhere, T by F , by T , T v and a v T by T , F v 8 and ’% v P by a, T & rll and ‘% & T by a, F & % and & F by F and by simultaneously dropping vacuous quantifiers, if necessary. Since all the mentioned pairs of S-formulas are pairs of equivalent 8-formulas, the 8-formula is transformed into an equivalent one. Since each reduction step diminishes the number
18
THE DECISION PROBLEM
of logical constants, this reduction can be carried on until T and F have disappeared from the S-formula, or until the S-formula is reduced to T or to F . This may be used to show that we need not investigate the validity or satisfiability of wffs containing propositional variables, because the propositional variables can be eliminated from a wff. Validity or satisfiability of a wff always means the validity of a certain wff containing no free variables. A WTJ with bound propositional variables may always be repluced by a n equivalent w g containing no propositional variables, or it m y be recognized at once as universally valid or valid in no domain at all. Each such wff can at first be transformed into an equivalent S-formula without bound propositional variables by replacing in the wff, beginning at an arbitrary point, every part ( X ) % ( X )by %(T)& %(F) and every part ( E X ) % ( X ) by %(T)v %(F). This S-formula is then reduced in the way described above. If the formula resulting in the end is T or F , the wff is universally valid or valid in no domain at all; otherwise, we get an equivalent wff without propositional variables. The well-known truth-table decision procedure for wffs of the propositional calculus, i.e. for wffs containing no individual and no predicate variables, is included herein. As an example we consider the wff:
( X ) ( Xv ( y ) ( F yv 15)) & ( E Y ) ( Y& (EG)(Ex)(Gz& Y ) ) . We first replace
( X ) ( X v (Y)(y=Y
" m by T v ( Y ) ( y = y v m
-
F v (Y)(Y=Y+
F)
which reduces to T,so that the wff itself is reduced to
(EY)(Y & (EG)(Ez)(Gz& Y ) ) . We replace this wff by
(T & (EG)(Ez)(Gx& T))v ( F & (EG)(Ez)(Gz& P ) ) which reduces to (EG)(Ex)Gx.(EG)(Ex)Gxis equivalent to the wff from which we were starting.
19
NORMAL FORMS
4.
Normal forms For this section we need the concept of P-constituent of a wff defined in 9 1 of Chapter I. A wff is said to express a tautology, or to be tautologous, if every S-formula reduces to T which results from the wff by replacing the P-constituents in an arbitrary way by T or F , but so that equally shaped P-constituents are replaced by the same letter. A tautologous wff is of course universally valid, but not every universally valid wff is tautologous. E.g. (z)(z= x) and (z)(Gzv&) are universally valid, but not tautologous. Two tautologous wffs are equivalent. Every wff can be brought into certain normal forms by means of equivalence transformations. There are several normal forms. A wff is said to be in conjunctive normal form if it is a disjunction the terms of which are P-constituents or the negations of P-constituents of the wff or if it is a conjunction of such disjunctions. Wffs which consist only of one P-constituent or of the negation of a P-constituent shall be included among the above disjunctions. We first show that for every non-tautologous wff an equivalent wff in conjunctive normal form can be constructed. a1,212, ..., am may be the P-constituents of the wff, P-constituents equally shaped not being counted twice. We will designate the wff by B(a1,..., a,). B(T, a2,..., a,) and B(F,212, ..., a,) are the #-formulas resulting from B(2Xl, ..., am)by replacing 2I1 everywhere by T ( F ) . B('2I1,..., a,) is equivalent to the S-formula
B(T,a2,..., %,). gets the value F , then the S-formula
a1v B(F, 212, ..., a,)
&,
v
For if by any assignment gets the same value by this assignment as
B v B ( F , a2,..., %), tk v B(T, a2,..., %,). This formula reduces to B(F, 212, ..., a,) which gets the same value
%(al,
as ..., a,). If by any assignment S-formula gets the same value as
a1 gets
the value T,the
T v B ( F , g2,..., '3%) & ?I v B(T,'?I2, ..., a,), which reduces to %(T,S2,..., a,).
20
THE DECISION PROBLEM
We shall prove our theorem by induction on the number n of P-constituents. If this number is one, %(9ll) is equivalent to v % ( F )& v %(T).We eliminate T and F in this 8-formula or 211 & % as conby the methods of 5 3 and get either 2117 or junctive normal form, since B(F)and %(T) cannot both reduce to T ;otherwise %(211) would be tautologous. Let us assume that we have shown our theorem to be true for a wff with less than n P-constituents. ..., a,) is equivalent to
a1
%(al,
211 v % ( F ,a2,..., a,) &
v
%(T,a2,...)am)
I n this S-formula F and T are again eliminated by the methods .of 0 3. The result is a wff of one of the following forms: 2l1v6&l,v%, 8 1 83%
v 9,
211v&&q, v 9,
a l V &
a1 &Z1, %I, &.
The wff cannot reduce to F , as is seen by its structure; it cannot reduce t o T,for in that case both %(F,a2,..., am) and %(T,g2,...,3%) would reduce to T,which is impossible since a2,..., is not tautologous. The last three of the above eight wffs are in conjunctive normal form. I n the other five wffs, 6 and 3 are wffs with only the P-constituents az7 ..., which according to our assumption can be brought into conjunctive normal forms 6’ and 9’. Because ‘ill1v G’ and v 9’can be transformed into a conjunctive normal form by means of the first equivalence E 2 1 of 5 2 , this holds too for the five wffs. This completes our induction. A tautologous wff with the P-constituents a17 ...)‘i&can also be given a conjunctive normal form; e.g. 3, v is such a form. A wff is in disjunctive normal form if it is a conjunction of P-constituents and the negations of P-constituents or if it is a disjunction of such conjunctions. Wffs consisting only of one P-constituent or of the negation of a P-constituent shall be included among the above conjunctions. I n order to show that a wff a can always be brought into a n equivalent disjunctive normal form, we fist construct a conjunctive normal form B1& ... & Brn of 8. According to E 2 of 5 2, 2l is equivalent to and therefore to B1& ... & 58%. Using E3,
%(al,
al
an)
21
NORMAL FORMS
Em.
is v ... v By E4 every !?jl& ... & !Bmis transformed into transformed into a conjunction whose terms are negations of P-constituents or double negated P-constituents. By eliminating the double negations according to E2, we get a disjunctive normal form of '21. Every wfl can be given an equivalent normal form in which the negation-bar stands only over prime formulas. If a wff is given, by successive applications of E3, E 4 , E5 and E6 the negation signs can be brought farther and farther inside until finally they stand, one or more times, only over the prime formulas. By means of E2 the negation signs over the prime formulas are removed until there is at most one left. Example: The wff
(G)(Ex)(Ey)(Gxy &2
7 )
becomes (EG)(x)(y)(= v x= y )
by the transformation. We m y further transform any w 8 into an equivalent one in which the quantifiers occur only at the beginning with scopes extending to the end of the w4, no quantifier standing under a negation-bar. We first transform the given wff into one in which the negationbars stand a t most over prime formulas. Then we rewrite the bound variables (see E l of 3 2 ) in a way such that all quantifiers have different variables. Subsequently, we place all the quantifiers at the beginning of the wff with scopes extending to the end of the wff, by using several times the equivalences E9-El6 of 5 2. This generally can be done in more than one way. For instance, the quantifiers can be placed a t the beginning of the wff in the order in which they occur, but also any succession of the initially placed quantifiers can be obtained such that any quantifier which was in the scope of another quantifier remains in the scope of that quantifier. A wff in which all the quantifiers are placed at the beginning of the wff with scopes extending to the end of the wff is said t o be in a prenex normal form. The succession of quantifiers with which the wff begins is called the prefix of the wff. The part of the wff which remains if all the quantifiers are deleted is called the matrix of the wff.
22
TI333 DECISION PROBLEM
Example: A prenex normal form of (x)(Ey)Gxy& (Ez)Hz is ( E x ) ( y ) ( E z ) ( K& y H z ) with the prefix ( E x ) ( y ) ( E zand ) the matrix & Hz. Other prenex normal forms are ( E x ) ( E z ) ( y ) ( K & y Hz) and ( E z ) ( E x ) ( y ) ( G& Hz). 5.
Three forms of the decision problem If a wff is valid (satisfiable) in a domain D, it is also valid (satisfiable) in a domain D, which has the same cardinal number as D,i.e. the elements of which may be put into one-to-one correspondence with those of D. Indeed, through the assignment used for the valuation of a wff in D,there may be determined in an unambigous way assignments relative to D, which give the same valuation. If a is assigned to an individual variable relative to D , the corresponding element a, of D, is used for the assignment relative to D,. If the n-adic propositional function rP is assigned to U" in D,the assignment for Unrelative to D,is the propositional function rP1 over Dlwhich assigns the value T to those, and only those, n-tuples (b,, ..., b,) over D,for which rP gives the value T to the corresponding n-tuples (a,, ..., a,) over D , a, corresponding t o bi. Therefore, disregarding those wfla which are universally valid (satisfiable in every non-empty domain) and disregarding those which have this property in no domain, the assertion of the validity (or satisfiability) of a w@ in a domain D is equivalent to a statement about the cardinal number of D. The definition of validity does not supply us with a criterion for recognizing in the general case whether a given wff is valid (satisfiable) in a given domain, or whether it is universally valid (satisfiable) at all. The problem of finding an effective procedure for determining the validity (satisfiability) of a given wff is called the decision problem. More precisely, as the validity and satisfiability of a wf€ depend on the cardinal number of the relative domain, the decision problem may be stated in several distinct forms. We formulate all of them with respect t o the problem of validity, since this is sufficient (see 8 1 of this chapter). The most important form is this:
THREE FORMS OF THE DECISION PROBLEM
23
I. T o decide for a given wfl whether it i s universally valid or not. A more comprising form of the problem is this: 11. To decide for a given wfl whether it is universally valid. If it i s not universally valid, to decide whether it i s valid in no domain whatsoever or in some domain. I n the last case the cardinal numbers of the domains for which it i s valid are to be determined. Other forms of the decision problem may be stated as well, e.g. the following one: 111. T o decide for a given wfl whether it i s valid in all domains with a finite number of elements or not. As far as the literature goes nearly all investigations about soluble cases of the decision problem concern problems I and 11; I11 has been dealt with only in connection with 11,since a solution of I1 includes a solution of I11 (but see Q 2 of chapter VIII). Of course it would be most desirable to have a decision procedure applicable to all wffs. Investigations made at first by A. Church [l, 21, then by A. Turing [l] and others, by which the somewhat vague intuitive notion of a decision procedure was replaced by a precise definition, show that a general solution of the decision problem in the forms I and I1 cannot be found. B. A. Trachtenbrod [l] has shown also that a general solution of the form I11 of the decision problem is not possible. To enter into the particulars of these investigations is outside the limits of this book. It remains t o look for a solution of the problem for special cases, i.e. for certain special well-defined classes of wffs. I n the following we give a survey of the more important special cases for which a solution has hitherto been reached.